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Topoi and Logic Appendix A Topoi and Logic In this section, we will explore the tight connection between topos theory and logic. In particular, to each topos there is associated a language for expressing the internal language of the topos. The converse is also true: given a language one can define a corresponding topos. A.1 First Order Languages A language, in its most raw definition, comprises a collection of atomic variables, and a collection of primitive operations called logical connectives, whose role is to combine together such primitive variables transforming them into formulas or sentences. Moreover, in order to reason with a given language, one also requires rules of inference, i.e. rules which allow you to generate other valid sentences from the given ones. The semantics or meaning of the logical connectives, however, is not given by the formulae and sentences themselves, but it is defined through a so called evaluation map, which is a map from the set of atomic variables and sentences to a set of truth values. Such a map enables one to determine when a formula is true and, thus, defines its semantics/meaning. In this perspective it turns out that the meaning of the logical connectives is given in terms of some set of objects which represent the truth values. The logic that a given language will exhibit will depend on what the set of truth values is considered to be. In fact, what was said above is a very abstract characterisation of what a language is. To actually use it as a deductive system of reasoning, one needs to define a mathematical context in which to represent this abstract language. In this way the elementary and compound propositions will be represented by certain mathematical objects and the set of truth values will itself be identified with an algebra. For example, in standard classical logic, the mathematical context used is Sets and the algebra of truth values is the Boolean algebra of subsets of a given set. C. Flori, A First Course in Topos Quantum Theory, Lecture Notes in Physics 868, 397 DOI 10.1007/978-3-642-35713-8, © Springer-Verlag Berlin Heidelberg 2013 398 A Topoi and Logic However, in a general topos, the internal logic/algebra will not be Boolean but will be a generalisation of it, namely a Heyting algebra. In order to get a better understanding of what a language is, we will start with a very simple language called propositional language P(l). A.2 Propositional Language The propositional language P(l) contains a set of symbols and a set of formation rules. Symbols of P(l) (i) An infinite list of symbols α0,α1,α2,...called primitive propositions. (ii) A set of symbols ¬, ∨, ∧, ⇒ which, for now, have no explicit meaning. (iii) Brackets ), (. Formation Rules (i) Each primitive proposition αi ∈ P(l)is a sentence. (ii) If α is a sentence, then so is ¬α. (iii) If α1 and α2 are sentences, then so are α1 ∧ α2, α1 ∨ α2 and α1 ⇒ α2. Note also that P(l) does not contain the quantifiers ∀ and ∃. This is because it is only a propositional language. To account for quantifiers one has to go to more complicated languages called higher-order languages, which will be described later. The inference rule present in P(l)is the modus ponens (the ‘rule of detachment’) which states that, from αi and αi ⇒ αj , the sentence αj may be derived. Symboli- cally this is written as αi,αi ⇒ αj . (A.1) αi We will see, later on, what exactly the above expression means. In order to use the language P(l) one needs to represent it in a mathematical context. The choice of such a context will depend on what type of system we want to reason about. For now we will consider a classical system, thus the mathematical context in which to represent the language P(l) will be Sets.InSets, the truth object (object in which the truth values lie) will be the Boolean set {0, 1}, thus the truth values will undergo a Boolean algebra. This, in turn, implies that the logic of the language P(l), as represented in Sets, will be Boolean. The rigorous definition of a representation of the language P(l)is as follows: Definition A.1 Given a language P(l)and a mathematical context τ , a representa- tion of P(l)in τ is a map π from the set of primitive propositions to elements in the algebra in question: α → π(α). A.2 Propositional Language 399 As we will see, the specification of the algebra will depend on what type of theory we are considering, i.e. classical or quantum. In classical physics, propositions are represented by the Boolean algebra of all (Borel) subsets of the classical state space, thus, given a representation π, we can define the semantics of P(l)as follows: π(αi ∨ αj ) := π(αi) ∨ π(αj ) π(αi ∧ αj ) := π(αi) ∧ π(αj ) (A.2) π(αi ⇒ αj ) := π(αi) ⇒ π(αj ) π(¬αi) := ¬ π(αi) where, on the left hand side, the symbols {¬, ∧, ∨, ⇒} are elements of the language P(l), while on the right hand side they are the logical connectives in algebra, in which the representation takes place. It is in such an algebra that the logical connec- tives acquire meaning. For the classical case, since the algebra in which a representation lives, is the Boolean algebra of subsets, the logical connectives on the right hand side of (A.2) are defined in terms of set-theoretic operations. In particular, we have the following associations: π(αi) ∨ π(αj ) := π(αi) ∪ π(αj ) (A.3) π(αi) ∧ π(αj ) := π(αi) ∩ π(αj ) c ¬ φ(αi) := π(αi) c π(αi) ⇒ π(αj ) := π(αi) ∪ π(αj ). (A.4) So far, we have seen how logical connectives are represented in the topos Sets. However, it is possible to give a general definition of logical connectives in terms of arrows. Such a definition would then be valid for any topos. To retrieve the logical connectives for the classical case, in which the topos is Sets, we then simply replace, in the definitions that will follow, the general truth object Ω with the set {0, 1}=2. Logical connectives in a general topos τ are defined as follows: • Negation We will now describe how to represent negation as an arrow in a given topos τ . Let us assume that the τ -arrow representing the value true is :1 → Ω, which is the arrow used in the definition of the sub-object classifier. Given such an arrow, negation is identified with the unique arrow ¬:Ω → Ω, such that the following 400 A Topoi and Logic diagram is a pullback: ⊥ 1 / Ω ¬ 1 / Ω Where ⊥ is the topos analogue of the arrow false in Sets,i.e.⊥ is the character of !1 : 0 → 1: !1 0 / 1 !1 ⊥ 1 / Ω • Conjunction Conjunction is identified with the following arrow: ∩:Ω × Ω → Ω which is the character of the product arrow , : 1 → Ω × Ω, such that the following diagram is a pullback: , 1 / Ω × Ω id1 ∩ 1 / Ω A.2 Propositional Language 401 where , is defined as follows: / q8 1 ΩO qqq I1 qq qq pr qqq 1 qq q , / 1 MM Ω × Ω MMM MMM MM pr2 MMM I1 M& 1 / Ω • Disjunction Disjunction is identified with the arrow ∪:Ω × Ω → Ω (A.5) which is the character of the image of the arrow , idΩ , idΩ , : Ω + Ω → Ω × Ω (A.6) such that the following diagram commutes: [ ,1Ω , 1Ω , ] Ω + Ω / Ω × Ω !Ω+Ω ∪ 1 / Ω • Implication Given two arrows ∩ / Ω × Ω / Ω pr1 their equaliser is some map e : ≤:= x,y|x ≤ y in Ω → Ω × Ω (A.7) ∩◦ = ◦ such that e pr1 e. Implication is then defined as the character of e,i.e.asthemap ⇒: Ω × Ω → Ω (A.8) 402 A Topoi and Logic such that the following diagram is a pullback: e ≤ / Ω × Ω !e ⇒ 1 / Ω In order to complete the definition of a propositional language in a general topos τ , we also need to define the valuation functions (which give us the semantics) in terms of arrows in that topos. We recall from the definition of the sub-object classifier that a truth value in a general topos τ is given by a map 1 → Ω (in Sets we have 1 →{0, 1}=2 = Ω). The collection of such arrows τ(1,Ω)represents the collection of all truth values. Thus, a valuation map in a general topos is defined to beamapV :{π(αi)}→τ(1,Ω)such that the following equalities hold: V ¬ π(αi) =¬◦V π(αi) (A.9) V π(αi) ∨ π(αj ) =∨◦ V π(αi) ,V π(αj ) (A.10) V π(αi) ∧ π(αj ) =∧◦ V π(αi) ,V π(αj ) (A.11) V π(αi) ⇒ π(αj ) =⇒◦V π(αi) ,V π(αj ) . (A.12) A.2.1 Example in Classical Physics We have stated above that classical physics uses the topos Sets. We now want to rep- resent, in Sets, the propositional language P(l), as defined for a classical system S. So now the elementary propositions will be propositions pertaining the physical sys- tem S. From now on, we will denote a language referred to a particular system S by P (l)(S). Since S is a (classical) physical system, the elementary propositions which P (l)(S) will contain will be of the form A ∈ Δ meaning “ the quantity A which represents some physical observable, has value in a set Δ”.
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